$g(r)=-(r+11)(r+14)$ 1) What are the zeros of the function? Write the smaller $r$ first, and the larger $r$ second. $\text{smaller }r=$
Answer: $\begin{aligned} -(r+11)(r+14)&=0 \\\\ r+11=0&\text{ or }r+14=0 \\\\ r={-11}&\text{ or }r={-14} \end{aligned}$ There are many ways to find the vertex. We will do it by using the fact that the $r$ -coordinate of the vertex is exactly between the two zeros. $\begin{aligned} \text{vertex's }r\text{-coordinate}&=\dfrac{({-11})+({-14})}{2} \\\\ &={-\dfrac{25}{2}} \end{aligned}$ Now we can find the vertex's $y$ -coordinate by evaluating $g\left({-\dfrac{25}{2}}\right)$ : $\begin{aligned} g\left({-\dfrac{25}{2}}\right)&=-\left({-\dfrac{25}{2}}+11\right)\left({-\dfrac{25}{2}}+14\right) \\\\ &=-\left(-\dfrac32\right)\left(\dfrac32\right) \\\\ &=\dfrac{9}{4} \end{aligned}$ In conclusion, $\begin{aligned} \text{smaller }r&=-14 \\\\ \text{larger }r&=-11 \end{aligned}$ The vertex of the parabola is at $\left(-\dfrac{25}{2},\dfrac{9}{4}\right)$